Adjusting for measurement error to quantify the relationship between diabetes and local access to healthy food

Ashley E. Mullan

Diabetes in Forsyth County

Health Outcome Mean Standard Error
Diabetes Prevalence 11.2% 3.3%
Lower Estimate 10.5% 3.2%
Upper Estimate 11.9% 3.5%


County Demographic Mean Standard Error
Population 4242 1581

Healthy Eating ➡️ Healthy Living

  • A healthy diet increases the likelihood of good overall health and decreases risk of preventable illness (World Health Organization, 2019).
  • Maintaining a healthy diet requires consistent access to healthy food, which may be hindered by geography or income.
  • Review studies found high prevalence of diabetes in food-insecure households (Gucciardi et al., 2014).

Choosing a Food Access Measure

  1. Count the number of healthy food retailers in a given radius (i.e., density)
  2. Compute the distance to the nearest healthy food retailer (i.e., proximity)
  3. Create an indicator of low food access that evaluates to \(1\) if zero healthy food retailers exist within a given distance

Distance Computations

  • The Haversine distance is a trigonometric function of latitude and longitude.
  • It ignores physical obstacles, so it underestimates the true distance between two points and is considered error-prone.
  • The Haversine distance in the image is impassable, as it crosses a pond.

Distance Computations

  • The route-based distance works around obstacles.
  • It is more accurate than the Haversine distance but is computationally expensive.
Distance Mean SE
Haversine 1.808 1.853
Route-Based 2.572 2.406
Difference 0.764 0.700

Guiding Questions

  1. Can we use a function of distance to healthy food retailers to quantify food access in Forsyth County, North Carolina, even if this function is subject to misclassification?

  2. Can we estimate the relationship between low food access and diabetes prevalence?

Two Phase Design

  • Having some correct route-based distances is better than none.
  • Error-prone Haversine distances are available for all \(N\) neighborhoods, and we can use them to create our indicator of low food access \(X^*\) that is subject to misclassification.
  • In addition to \(X^*\), we query route-based distances \(X\) for \(n\) neighborhoods, where \(n < N\).

Existing Methods

  • Functions of sensitivity and specificity can be used to estimate prevalence of an imperfectly classified outcome (Speybroeck et al., 2013).

  • Binary variables may be created from continuous ones with non-differential error, but the new variables may still have differential error (Flegal et al., 1991).

  • Validation data can be used to build a model for \(X\) and use it for multiple imputation (Cole et al., 2006).

Existing Methods

  • Naive regression models, where error-prone \(X^∗\) is used in the place of \(X\) will have regression coefficients that are biased by a function of the sensitivity and specificity (Shaw et al., 2020).

  • We follow Tang et al., and others by considering a maximum likelihood estimator (MLE) that incorporates both queried and unqueried observations.

Simulation Roadmap

We compare the gold standard approach to the following approaches:

  1. Naive Analysis (replace \(X\) with \(X^*\))
  2. Complete Case Analysis (only use rows that have \(X\))

The coefficient we are looking to estimate has true value \(\beta_1 = 2\) throughout the simulation study.

Naive Analysis Simulations

Complete Case Simulations

Simulation Summary

  • The naive approach is biased when compared to the gold standard.
  • The complete-case approach is unbiased but has a higher variance than the gold standard and is inefficient.
  • We want to our estimator be unbiased and efficient.

Notation

  • \(X\) is an error-free binary explanatory variable
  • \(X^*\) is an error-prone version of \(X\)
  • \(\boldsymbol{Z}\) is an error-free covariate vector
  • \(Y\) is an outcome (diabetes prevalence in a neighborhood)

We are interested in finding the coefficient vector \(\boldsymbol{\beta}\) from the Poisson model of \(Y \mid X, \boldsymbol Z\).

Building the Log-Likelihood

The log-likelihood function is constructed based on the available data, and we build up from the ideal to reality through the following three cases.

  1. Accurate and Not Missing 🤩
  2. Not Accurate but Not Missing 🤨
  3. Not Accurate and Missing 🤔

Simple Case (🤩)

  • All observations take the form \((X,Y,\boldsymbol{Z})\).
  • Our Poisson model of interest comes from breaking up the joint probability of an observation.

\[P(Y,X,\boldsymbol{Z}) = P_\boldsymbol{\beta}(Y \mid X, \boldsymbol{Z})P(X \mid \boldsymbol{Z})P(Z)\]

  • We end up maximizing: \(\sum\limits_{i = 1}^{N}\log\{P_{\boldsymbol{\beta}}(Y_i \mid X_i, \boldsymbol{Z}_i)\}\)

The Useless Case (🤨)

  • All observations take the form \((X, X^*, Y, \boldsymbol{Z})\).
  • Again, break up the joint probability and simplify.

\[P_{\boldsymbol{\beta}}(Y,X,\boldsymbol{Z}, X^*) \propto P_{\boldsymbol{\beta}}(Y \mid X, \boldsymbol{Z})P(X \mid X^*, \boldsymbol{Z}).\]

  • We don’t care about this case, as it is not necessary to use \(X^*\) if we have fully observed \(X\), and we end up maximizing the same function as the ideal case.

Our Case (🤔)

  • We have employed two-phase design to ensure that \(n\) observations take the form \((X, X^*, Y, \boldsymbol{Z})\) and \(N-n\) observations take the form \((X^*, Y, \boldsymbol{Z})\).
  • Using surrogacy, the conditional independence of \(X^*\) and \(Y\) given \(X\), we can set up the outcome model and the error model, parameterized by \(\boldsymbol{\beta}\) and \(\boldsymbol{\eta}\) respectively.

Our Case (🤔)

We first handle the \(n\) queried observations.

\(P(Y,X,\boldsymbol{Z}, X^*)\)

\(\phantom{PPPPP} = P_{\boldsymbol{\beta}}(Y \mid X, X^*, \boldsymbol{Z})P_{\boldsymbol{\eta}}(X \mid X^*, \boldsymbol{Z})P(X^*, \boldsymbol{Z})\) \(\phantom{PPPPP} = P_{\boldsymbol{\beta}}(Y \mid X, \boldsymbol{Z})P_{\boldsymbol{\eta}}(X \mid X^*, \boldsymbol{Z})P(X^*, \boldsymbol{Z})\)

We now handle the other \(N-n\) observations by marginalizing over \(X\).

\[P(Y,X^*,\boldsymbol{Z}) = \sum\limits_{x=0}^{1}P(Y,X=x,\boldsymbol{Z}, X^*)\]

Our Final Log Likelihood

  • In the full log likelihood, each observation’s contribution is additive, so we can now use all \(N\) observations along with \(Q_i\), an indicator for the query status of observation \(i\).

  • We are ready to maximize the equation below using numerical methods in \(R\) to solve for \(\hat{\boldsymbol{\beta}}\) and better understand the relationship of interest.

\[\ell(\boldsymbol{\beta}, \boldsymbol{\eta}) = \sum\limits_{i = 1}^N Q_i\log P_{\boldsymbol{\beta},\boldsymbol{\eta}}(X,X^*,Y, \boldsymbol{Z}) + \sum\limits_{i = 1}^N (1 - Q_i)\log P_{\boldsymbol{\beta},\boldsymbol{\eta}}(Y, X^*, \boldsymbol{Z}).\]

Next Steps

  1. Implement this MLE in an R package.
  2. Conduct simulations to assess its bias and efficiency.
  3. Apply this estimator to our real data.

References

S. R. Cole, H. Chu, and S. Greenland. Multiple-imputation for measurement-error correction. International Journal of Epidemiology, 35(4):1074–1081, 2006.

K. M. Flegal, P. M. Keyl, and F. J. Nieto. Differential misclassification arising from nondifferential errors in exposure measurement. American Journal of Epidemiology, 134(10):1233–1246, 1991.

E. Gucciardi, M. Vahabi, N. Norris, J. P. Del Monte, and C. Farnum. The intersection between food insecurity and diabetes: a review. Current nutrition reports, 3:324–332, 2014.

World Health Organization. Healthy diet, 2019. URL https://iris.who.int/handle/10665/325828.

References

P. A. Shaw, R. H. Keogh, et al. STRATOS guidance document on measurement error and misclassification of variables in observational epidemiology: part 2—more complex methods of adjustment and advanced topics. Statistics in medicine, 39(16):2232–2263, 2020.

B. E. Shepherd, P. A. Shaw, and L. E. Dodd. Using audit information to adjust parameter estimates for data errors in clinical trials. Clinical Trials, 9(6):721–729, 2012.

N. Speybroeck, B. Devleesschauwer, L. Joseph, and D. Berkvens. Misclassification errors in prevalence estimation: Bayesian handling with care. International journal of public health, 58:791–795, 2013.

L. Tang, R. H. Lyles, C. C. King, D. D. Celentano, and Y. Lo. Binary regression with differentially misclassified response and exposure variables. Statistics in Medicine, 34(9):1605–1620, 2015.

Thank You!

  • Dr. Lotspeich for her guidance
  • The SESH Lab for their advice on this talk
  • Dr. McGowan for her Quarto skills
  • The audience for coming today!!!

STA791: Fall 2023